Physical systems are conventionally studied using experiments, that may be costly, exceedingly difficult, or infeasible. Alternatively, various computational approaches, have proven rigorous and successful, but still suffer from being generally resource-intensive and occasionally difficult to interpret. Particularly for classes of systems that can be described by partial differential equations, such as for example multi-component crystalline solids undergoing mechanical deformations, and changes in chemical potential, high-fidelity solutions generally contain up to tens of millions of degrees of freedom. New methods must, therefore, be developed that represent this information in a lower-dimensional space, allowing for more efficient, and intuitive computations.

In this talk, I will be introducing a novel graph-theoretic approach for reduced-order modelling of a wide class of systems, allowing for more efficient and effective data-driven simulations. Concepts in graph theory will be introduced, including a rigorous non-local discrete calculus, and I will describe how states of a system and their relationships can be represented using this formalism. I will then discuss the numerical methods for computing a reduced-order model for quantities of interest and will show some preliminary results, indicating the validity and possible exciting future applications of this general framework.